Investigate the number of winning lines in the game of connect 4.

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Investigation

Connect 4

Task

This is a winning line in the game of connect 4 on a 4x5 board. Winning lines can be horizontal, vertical and diagonal. Investigate the number of winning lines in the game of connect 4.

The task is asking me to find out how many winning lines (connects) when you are connecting 4 there are on any size board.

What am I going to do

I am going to find out how many connect 4 there are in a 4x5 board.

•I will change the size of the box, but keep one value the width constant. And I will find a pattern in the number of connects there are in the different size boxes.

•I will use algebra to find a general formula for a NxWidth (W) box.

•I will then increase the width (constant) by one and work out a formula for that box.

•I will then find a pattern in the formulas for the different size boxes, connecting 4, and I will make a formula for the formula.

•I will then change the number that I will connect. For example 2, 3 or 5.

Connect 4

Firstly I will do a box with the width constant as 5 and I will change the height.    

                                                 

                         Hx5 Box

Any Number=N

Connects=C

Height= H

Width =W

                    Hx5   1   2   3   4    5    6        

            Connects   2   4   6  17  28  39          first layer

                                         11  11  11             second layer                                                            

The box height of 1 and 2 do follow the pattern so

they are excluded. The connects go up by 11 each

time.   There are only 2 layers so the equation we

use is this equation. C=aH+b (original equation)

C=aH+b.

We put in results from the above table and since the

connects has 2 layers we need to put in results

twice.

  6=a3+b       (1)

17=a4+b       (2)

        11=a        (2)-(1)

Substitute ‘a’ which is 11 back into (2)

17=11x4+b

17=44+b

                                         b=17- 44      b = 27

        b=-27 substitute ‘a’ and ‘b’ back into original  

        equation.

        C=11H-27    that is the equation for the number of

                        connects in a Nx5 box. But since the first 2 heights

                        didn’t follow the pattern we didn’t use them in the

                        equation so this equation doesn’t work for them.

Connect 4

               Hx6 Box

        

        Hx6   1   2   3   4   5   6

      Connects   3   6   9   24 39 54         first layer

                                    15  15  15          second layer

As before the first 2 equations do not follow  

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the pattern so they are excluded. Also like before  

the connects has 2 layers so we use this original

equation.

C=aH+b

We use the same method as before.

  9=a3+b     (1)

24=a4+b     (2)

15=a           (2)-(1)

Substitute ‘a’ which is 15 back into (1)

9=15x3+b

9=45+b

                           b= -36

Substitute ‘a’ and ‘b’ back into original equation

C=15H-36    that is the equation for the number  

of connects in a Nx6 box. But since the            

      ...

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